From 7a3c35fa648374f0f18181cd73588f1febbdcb93 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 8 Dec 2020 22:11:56 +0100
Subject: Added some knarly equations.

---
 bezout.tex | 69 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++-----
 1 file changed, 64 insertions(+), 5 deletions(-)

diff --git a/bezout.tex b/bezout.tex
index 5ea3a33..bdb7dd4 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -124,7 +124,7 @@ or the norm squared of that of an $N\times N$ complex symmetric matrix.
 These equivalences belie a deeper connection between the spectra of the
 corresponding matrices: each eigenvalue of the real matrix has a negative
 partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ is an
-eigenvalue of the Hermitian matrix. Finally, $|\lambda|$ is a \emph{singular
+eigenvalue of the Hermitian matrix and $|\lambda|$ is a \emph{singular
 value} of the complex symmetric matrix. The distribution of positive
 eigenvalues of the Hessian is therefore the same as the distribution of
 singular values of $\partial\partial H$, while both are the same as the
@@ -140,7 +140,56 @@ study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{
 A useful property of the Gaussian distributions is that gradient and Hessian
 may be seen to be independent \cite{Bray_2007_Statistics,
 Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the
-Hessians as independent. We compute each by taking the saddle point.
+Hessians as independent. We compute each by taking the saddle point. The
+$\delta$-functions are converted to exponentials by the introduction of
+auxiliary fields $\hat z=\hat x+i\hat y$. The average over $J$ can then be
+performed. A generalized Hubbard--Stratonovich then allows a change of
+variables from the $4N$ original and auxiliary fields to eight bilinears
+defined by
+\begin{equation}
+  \begin{aligned}
+    Na=z^*\cdot z
+    &&
+    N\hat c=\hat z\cdot\hat z
+    &&
+    Nb=\hat z^*\cdot z \\
+    N\hat a=\hat z^*\cdot\hat z
+    &&
+    Nd=\hat z\cdot z
+  \end{aligned}
+\end{equation}
+and their conjugates. The result is, to leading order in $N$,
+\begin{equation} \label{eq:saddle}
+    \overline{\mathcal N_J}(\kappa,\epsilon)
+        = \int da\,d\hat a\,db\,db^*d\hat c\,d\hat c^*dd\,dd^*e^{Nf(a,\hat a,b,\hat c,d)},
+\end{equation}
+where
+\begin{widetext}
+  \textcolor{red}{\textbf{[appendix?? I'm putting too much right now so as to trim later...]}}
+  \begin{equation}
+    f=2+\frac12\log\det\frac12\begin{bmatrix}
+      1 & a & d & b \\
+      a & 1 & b^* & d^* \\
+      d & b^* & \hat c & \hat a \\
+      b & d^* & \hat a & \hat c^*
+    \end{bmatrix}
+    +\mathop{\mathrm{Re}}\left\{\frac18\left[\hat aa^{p-1}+(p-1)|d|^2a^{p-2}+\kappa(\hat c^*+(p-1)b^2)\right]-\epsilon b\right\}
+    +\int d\lambda\,\rho(\lambda)\log|\lambda|^2
+  \end{equation}
+  where $\rho(\lambda)$, the distribution of eigenvalues $\lambda$ of
+  $\partial\partial H$, is dependant on $a$ alone. This function has a maximum in
+  $\hat a$, $b$, $\hat c$, and $d$ at which its value is (for simplicity, with
+  $\kappa\in\mathbb R$)
+  \begin{equation}
+    \begin{aligned}
+      f(a)&=1+\frac12\log\left(\frac4{p^2}\frac{a^2-1}{a^{2(p-1)}-\kappa^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 \\
+          &\hspace{80pt}-\frac{a^p(1+p(a^2-1))-a^2\kappa}{a^{2p}+a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Re}}\epsilon)^2
+          -\frac{a^p(1+p(a^2-1))+a^2\kappa}{a^{2p}-a^p(a^2-1)(p-1)-a^2\kappa^2}(\mathop{\mathrm{Im}}\epsilon)^2,
+    \end{aligned}
+  \end{equation}
+\end{widetext}
+This leaves a single parameter, $a$, which dictates the magnitude of $z^*\cdot z$, or alternatively the magnitude $y^2$ of the imaginary part. The latter vanishes as $a\to1$, where we should recover known results for the real $p$-spin.
+
 
 The Hessian of \eqref{eq:constrained.hamiltonian} is $\partial\partial
 H=\partial\partial H_0-p\epsilon I$, or the Hessian of
@@ -168,7 +217,7 @@ where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
 spectrum of $\partial\partial H$ therefore is that of an ellipse whose center
 is shifted by $p\epsilon$.
 
-\begin{figure}
+\begin{figure}[htpb]
   \centering
 
   \raisebox{60pt}{$|\epsilon|=0$}
@@ -210,8 +259,18 @@ singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the
 elements of $J$ are standard complex normal, this corresponds to a complex
 Wishart distribution. For $\kappa\neq0$ the problem changes, and to our
 knowledge a closed form is not known.  We have worked out an implicit form for
-this spectrum using the saddle point of a replica calculation for the Green
-function. blah blah blah\dots
+this spectrum using the saddle point of a replica symmetric calculation for the
+Green function. The result is
+\begin{widetext}
+  \begin{equation}
+    G(\sigma)=\lim_{n\to0}\int d\alpha\,d\chi\,d\chi^*\frac\alpha2
+    \exp nN\left\{
+      1+\frac{p(p-1)}{16}a^{p-2}\alpha^2-\frac{\alpha\sigma}2+\frac12\log(\alpha^2-|\chi|^2)
+      +\frac p4\mathop{\mathrm{Re}}\left(\frac{(p-1)}8\kappa^*\chi^2-\epsilon^*\chi\right)
+      \right\}
+  \end{equation}
+\end{widetext}
+The argument of the exponential has several saddles, but the one with the smallest value of $\mathop{\mathrm{Re}}\alpha$ gives the correct solution \textcolor{red}{\textbf{why????? we never figured this out...}}.
 
 The transition from a one-cut to two-cut singular value spectrum naturally
 corresponds to the origin leaving the support of the eigenvalue spectrum.
-- 
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